<p>We study the boundedness of wave operators associated with one-dimensional Schrödinger operators with a generalized point interaction. Let <i>H</i> denote the Laplacian with a generalized point interaction at the origin and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> the free Hamiltonian in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2({\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that the wave operators <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$W_\pm (H,H_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mo>±</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are extended to bounded operators on the Sobolev spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$W^{k,p}(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$k=0,1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. The boundedness of wave operators has been established for Schrödinger operators with decaying potentials by Yajima [<CitationRef CitationID="CR25">25</CitationRef>, <CitationRef CitationID="CR26">26</CitationRef>] and Weder [<CitationRef CitationID="CR24">24</CitationRef>] and for certain singular perturbations such as <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-interactions by Duchêne et al. [<CitationRef CitationID="CR9">9</CitationRef>]. The present result shows that the boundedness extends to the whole class of generalized point interactions.</p>

The $W^{1,p}$-continuity of the wave operators for the one-dimensional Schrödinger operators with a generalized point interaction

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发表日期：

2026年7月30日

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Hiroaki Niikuni

摘要

We study the boundedness of wave operators associated with one-dimensional Schrödinger operators with a generalized point interaction. Let H denote the Laplacian with a generalized point interaction at the origin and $H_0$ $H_{0}$ the free Hamiltonian in $L^2({\mathbb {R}})$ $L^{2} (R)$ . In this paper, we prove that the wave operators $W_\pm (H,H_0)$ $W_{\pm} (H, H_{0})$ are extended to bounded operators on the Sobolev spaces $W^{k,p}(\mathbb {R})$ $W^{k, p} (R)$ for $k=0,1$ $k = 0, 1$ and $1<p<\infty $ $1 < p < \infty$ . The boundedness of wave operators has been established for Schrödinger operators with decaying potentials by Yajima [25, 26] and Weder [24] and for certain singular perturbations such as $\delta $ $δ$ -interactions by Duchêne et al. [9]. The present result shows that the boundedness extends to the whole class of generalized point interactions.

The \(W^{1,p}\)-continuity of the wave operators for the one-dimensional Schrödinger operators with a generalized point interaction

摘要